Notation:

We say two $C^1$ manifolds are $C^1$-homeomorphic if they are homeomorphic via a $C^1$ homeomorphism with $C^1$ inverse.

Question:

Let $n \geq 2$. Given a countable dense set of points $P \subset \mathbb R^n$, does there exist a $C^1$ foliation $M_{\alpha}$ of $\mathbb R^n$ by $C^1$ manifolds that are $C^1$-homeomorphic to $\mathbb R^{n-1}$ with the following property?

Suppose the foliation $M_{\alpha}$ is parametrised by $\alpha \in (0, 1)$. Then the set of all $\alpha$ such that $M_{\alpha} \cap P$ is dense in $M_{\alpha}$ is dense in $(0, 1)$.